A Hamilton–Jacobi variational framework for nonlinear control of renewable energy conversion systems

Abstract

Wind and photovoltaic grid interfaces belong to a family of renewable energy conversion systems which are characterized by strongly coupled nonlinear dynamical systems, and as such cannot be effectively controlled using techniques such as H infinity and linear-quadratic-regulator which only yield local stability guarantees. The authors of this paper propose a Hamilton-Jacobi optimization framework in which the problem of nonlinear renewable energy control is expressed as a zero-sum minimax game happening between control action and the disturbance, leading to a specific form of the value function represented via the viscosity solution of the Hamilton Jacobi-Bellman-Isaacs (HJBI) partial differential equation. In the paper, the authors introduce the standard Lipschitz, linear-growth, and convexity assumptions for the dynamics and costs, show local Lipschitz continuity and monotonicity of the Hamiltonian, prove the comparison principle for bounded sub-and super-solutions, establish existence and uniqueness of viscosity solution through the context of Perron-type construction, synthesize state-feedback controller from value function generalized gradient, analyze closed-loop stability using value function as a Lyapunov function and introducing a semi-discrete Lax-Friedrichs/ENO scheme proving its uniform convergence to the viscosity solution. It has been demonstrated how the framework performs on a synchronous generator-converter example with three-phase fault and discretization on grids from 0.2 to 0.00625 state spacing. The Lax-Friedrichs scheme has shown first-order approach to convergence with observed value of 1.02, while ENO scheme performs close to second-order with 1.85 of the observed value. Compared to H infinity, Linear Quadratic-Regulator, and backstepping sliding mode baselines, the synthesized controller demonstrates a reduction of the normalized worst-case cost by 29.6-40.5% and the settling time by 23.5-47.5%. These results indicate that the proposed variational framework yields a mathematically well-posed and numerically verifiable nonlinear controller with measurable robustness advantages for renewable energy conversion systems operating under large disturbances.